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%I #2 Mar 30 2012 18:37:16
%S 1,1,0,-7,0,242,0,-17771,0,2189294,0,-404590470,0,104785114020,0,
%T -36267349272243,0,16185534555969878,0,-9056084951164614194,0,
%U 6210575966204050489916,0,-5124228589785627978616702,0
%N G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x/F(x)^2) have zeros for every other coefficient after initial terms; g.f. of dual sequence A157304 satisfies the same condition.
%F For n>=1, [x^(2n+1)] 1/A(x)^(4n+1) = 0.
%F G.f. satisfies: A(x) = F(x*A(x)^2) where F(x) = A(x/F(x)^2) = sqrt(x/Series_Reversion(x*A(x)^2)) = g.f. of A157305.
%F G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = x/Series_Reversion(x*A(x)) = g.f. of A157306.
%e G.f.: A(x) = 1 + x - 7*x^3 + 242*x^5 - 17771*x^7 + 2189294*x^9 -+...
%e ...
%e Let F(x) = A(x/F(x)^2) so that A(x) = F(x*A(x)^2) then
%e F(x) = 1 + x - 2*x^2 + 26*x^4 - 1378*x^6 + 141202*x^8 -+...
%e has alternating zeros in the coefficients (cf. A157305):
%e [1,1,-2,0,26,0,-1378,0,141202,0,-22716418,0,5218302090,0,...].
%e ...
%e COEFFICIENTS IN ODD NEGATIVE POWERS OF G.F. A(x).
%e A^1 : [(1),1,0,-7,0,242,0,-17771,0,2189294,0,-404590470,0,...];
%e A^-1: [1,(-1),1,6,-13,-222,506,16932,-37709,-2127126,4595294,...];
%e A^-3: [1,-3,(6),11,-69,-537,2806,45282,-215781,-5963673,...];
%e A^-5: [1,-5,15,(0),-140,-601,6245,62380,-503935,-8911515,...];
%e A^-7: [1,-7,28,-35,(-182),-392,9968,65519,-860825,-10670499,...];
%e A^-9: [1,-9,45,-102,-135,(0),13128,54504,-1240416,-11070241,...];
%e A^-11:[1,-11,66,-209,77,341,(15158),31460,-1598696,-10074240,...];
%e A^-13:[1,-13,91,-364,546,221,16107,(0),-1899508,-7767240,...];
%e A^-15:[1,-15,120,-575,1380,-978,17040,-36375,(-2118030),...];
%e A^-17:[1,-17,153,-850,2703,-4114,20502,-76772,-2240175,(0),...];
%e ...
%e When scaled, the coefficients shown above in parenthesis
%e forms the coefficients of the function F(x) = A(x/F(x)^2):
%e F: [1,-1/(-1),6/(-3),0,-182/(-7),0,15158/(-11),0,-2118030/(-15),0,...].
%o (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); Vec(serreverse(x/Ser(A))/x)[n+1]}
%Y Cf. A157305, A157306, A157302, A157303, A157304 (dual).
%K sign
%O 0,4
%A _Paul D. Hanna_, Feb 28 2009
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